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Some Stability Inequalities for Compact Finite Difference Schemes

✍ Scribed by Rolf Dieter Grigorieff

Publisher: John Wiley and Sons
Year: 1988
Tongue: English
Weight: 460 KB
Volume: 135
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.19881350110

No coin nor oath required. For personal study only.

✦ Synopsis

For finite difference schemes of compact forni on noniiniform grids approximating w t h order two-point boiindary value problenia stabilit!-inequulitiea are proved irliicll use ti norm anitlogotis t o the Gprncm-norm in the ease of milltistep niethocts. TIM restilts are appliecl to a nitmber of finite difference sriirnies for which they establish a higher o d e r of convergence t11m naively esperted.

1. Introduction.

Invmtipting the hehavior of multistep methods for ordinary initin1 d u e problems SPr.ncm [TI introduced for grid functions ?th the norm li%hli:=mns {!?&(zo)+ h % h ( r k ) : . i=o, ..., N ) k 4

nowadays colled the SPI.JKLR norm. which he showed to be IL useful tool in order to estnblish p c i s e cmvergence results for schemes behnving rather errstic. The purpose of this p q e r is to pmve similar stability inequalities for finite difference schemes of m m p t form on nonuniform grids approximating tao-point boundary value problems or mote genenlly certain integrodifferentiall equ a t' ions.

The main theorem is (14) in section 3 from which, under simplifying conditions,

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